ETA: I wanted to add that your question was about email etiquette, and it is hard to know. Just because the email address is on a website, doesn't necessarily mean it is intended to be used for anything beyond business. Is it a personal email or a business email? If it is a business email, does it go directly to him, or is it filtered perhaps through a secretary/office person?

If it is a personal email (specifically to the therapist), it still is something to discuss with the therapist. Some therapist only want email used for business matters. Others don't mind emails but don't reply. Others reply regularly. It helps to know what the guidelines are and since you don't, you might should discuss it first just so you are on the same page.

Different, but not that different. My cardiologist has an email on his website; it looks like it would go straight to him, but apparently it was more of a business email that he honestly didn't use much, pretty much read and filtered through office personnel. However, he personally handed me a different email address that he wanted me to use to correspond directly with him about my condition and treatment. He told me specifically what he wanted me to use it for, that this was the email he wanted me to use, and what I could expect as a far as timing of replies, etc. Because he was so specifically about it, I knew exactly when and how to use it and what to expect in return. (It was actually very personal, helpful, and conversational dialogue via email which was rather amazing coming from a medical specialist.)

Similarly, my husband's pain specialist provided me with his personal cell phone number. I understand it is only to be used in cases of extreme emergency if his answering service doesn't handle matters correctly (which was what we had run into). We know those guidelines ahead of time and therefore know the etiquette for that number's use.